Kraft made no secret that he believed his young prodigy, publicly heralding his self-proclaimed maturity.

But now Hernandez’s claims for himself and the authenticity of the so-called Patriots’ Way have fallen under scrutiny as State Police homicide investigators examine Hernandez’s possible link to the shooting death of his acquaintance, Odin Lloyd.

‘I’ve got this thing. You only hang out with good people. You get the turkeys out of your life.’

Police converged on Hernandez’s North Attleborough home after Lloyd, 27, of Dorchester, was discovered dead Monday in an industrial park less than a mile from Hernandez’s home. No suspects have been identified in the murder.

AP FILE PHOTO

Aaron Hernandez hadn’t had any major behavioral problems with the Patriots.

Hernandez, 23, is the latest in a string of Patriots players who have become entangled in police investigations in recent years. Amid the fallout, the proud image Kraft has projected — that his franchise is an institution for men of respectable character — has never been more in question.

A spokesman for Kraft and the Patriots said they declined to comment.

In 2011, Kraft told the Associated Press, “In my family, we treat everyone the same, as long as they are people of character. I’ve got this thing. You only hang out with good people. You get the turkeys out of your life.’’

Last year alone, the Patriots reacquired wide receiver Donté Stallworth, who killed a man while driving drunk in 2009, and cornerback Aqib Talib, who was charged in 2009 with assaulting a taxi driver and indicted in 2011 for allegedly firing a gun at his sister’s boyfriend.

With a proud and proven winning tradition, the 21st century Patriots have tried with mixed results to rehabilitate some gifted players who have failed, sometimes badly, off the field.

The list includes defensive back Alfonzo Dennard, who was charged with punching a police officer in Nebraska less than a week before the Patriots selected him in last year’s NFL draft. Dennard was convicted in April of assault and sentenced to serve 30 days in jail next March.

In 2011, the Patriots acquired defensive tackle Albert Haynesworth, five months after he was charged with punching a motorist in the nose during a road-rage incident.

Three years earlier, Haynesworth allegedly was driving his Ferrari more than 100 miles an hour when he crashed and left another motorist partially paralyzed.

Just weeks after the Patriots signed linebacker Brandon Spikes to a four-year contract in 2010, his image went viral as the star of an Internet pornography video. Spikes later was suspended four games for violating the NFL’s substance abuse policy.

In 2006, the Patriots drafted safety Willie Andrews, four years after he was sentenced to 30 days in jail for a gun possession conviction.

He played in the 2008 Super Bowl for the Patriots, and two days later was arrested with a half-pound of marijuana and a large sum of money in his unregistered car.

Even then, the Patriots retained Andrews until he was arrested five months later for allegedly drawing a gun on his girlfriend.

A star running back, Corey Dillon, who helped the Patriots win a Super Bowl in 2005, had been charged in 2000 with assaulting his wife while he was a member of the Cincinnati Bengals.

There also were drug cases involving Nick Kaczur (oxycodone in 2008) and Kevin Faulk (marijuana possession in 2008) while they were Patriots.

Each transgression has tarnished Kraft’s record of trying to win Super Bowls with teams of admirable character. Yet the Patriots should not necessarily be denigrated for their history of player-related legal woes, said Dan Lebowitz, executive director of the Center for Sport in Society at Northeastern University.

He said American culture is rife with men behaving badly. That some are pro athletes should come as no surprise.

“I understand that there used to be a Patriot Way and there was a lot of talk about the Krafts running a family business,’’ Lebowitz said. “I think in many respects they probably still do. But it’s a fine balance between winning and finding the right kind of players they want to help them do that.’’

The Patriots made a powerful statement about the type of problem players they would shun after Kraft bought the franchise in 1994. Two years later, the team drafted defensive tackle Christian Peter, only to have Kraft take the uncommon step of relinquishing team rights to the player when he discovered Peter’s background was more unsavory than he believed.

Peter had a history of violence against women.

“I don’t want thugs and hoodlums here,’’ Kraft told Bill Parcells, his coach at the time, according to the 2012 book, “Coaching Confidential: Inside the Fraternity of NFL Coaches.’’

Hernandez’s dubious past, including a positive marijuana test in college and his long association with “street activity’’ in his hometown of Bristol, Conn., raised concerns about his character among some NFL executives before the 2010 draft, the Globe reported Friday.

Those concerns caused Hernandez to fall to the fourth round of the draft as every other NFL team passed him over before the Patriots selected him, 113th overall, out of the University of Florida.

Hernandez had no history of major behavioral problems with the Patriots until news surfaced this week that Alexander Bradley, 30, who has described himself as Hernandez’s former paid assistant, is suing Hernandez for allegedly shooting him in the face Feb. 13 during an argument after they left a Miami strip club, using a gun Hernandez possessed illegally.

Police investigated the case, but no criminal charges have been filed. Bradley, who has served prison time in Connecticut for a cocaine conviction, is suing Hernandez in federal court for more than $100,000 because of the February shooting.

While the Patriots wait for additional information about Hernandez in both the Miami and North Attleborough cases, Lebowitz said the team can take a measure of pride from recently acquiring Tim Tebow, an evangelical Christian whose off-field behavior is unlikely to further embarrass Kraft’s franchise.

Shalise Manza Young of the Globe staff contributed to this report. Bob Hohler can be reached at hohler@globe.com.